The effects of small vibrations on a particle oscillating near a solid wall in a fluid cell, relevant to material processing such as
crystal growth in space, have been investigated by direct numerical simulations. Simulations have been conducted for a solid
particle suspended in a fluid cell filled with a fluid of 1 cSt viscosity, vibrating sinusoidally in a horizontal direction. The
simulations revealed the existence of a vibration-induced force attracting the particle towards the nearest cell wall which
varied with the cell vibration frequency. The predicted flow patterns around the particle revealed that different velocity and
pressure distributions would be induced by the particle motion. In particular, the flow in the gap between the particle and the
nearest wall was predicted to accelerate and pressure to decrease in accordance with Bernoulli’s principle, which would result
in the attraction force.

General Note:

The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows

Vibration-induced attraction of a particle from a wall in microgravity

Mehrrad Saadatmanda, Masahiro Kawajia'* and Howard H. Hub

'Dept. of Chemical Engineering and Applied Chemistry, University of Toronto, Toronto, ON M5S 3E5, Canada
bDept. of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA, USA

The effects of small vibrations on a particle oscillating near a solid wall in a fluid cell, relevant to material processing such as
crystal growth in space, have been investigated by direct numerical simulations. Simulations have been conducted for a solid
particle suspended in a fluid cell filled with a fluid of 1 cSt viscosity, vibrating sinusoidally in a horizontal direction. The
simulations revealed the existence of a vibration-induced force attracting the particle towards the nearest cell wall which
varied with the cell vibration frequency. The predicted flow patterns around the particle revealed that different velocity and
pressure distributions would be induced by the particle motion. In particular, the flow in the gap between the particle and the
nearest wall was predicted to accelerate and pressure to decrease in accordance with Bernoulli's principle, which would result
in the attraction force.

Introduction

The space platforms with near zero gravity environment
have been considered an ideal place for the production of
high quality protein and semiconductor crystals from
solutions or melts, by eliminating the sedimentation effect
and buoyancy-induced convective motion of the fluid
experienced on Earth. Ideally, in the absence of gravity,
diffusive mass transfer would dominate the crystal growth
which can result in the formation of crystals of better
morphologies or properties. Motivated by this idea, there
have been many experiments conducted aboard space
platforms to produce, for example, protein crystals of better
quality. The results, however, have shown unexpected
deviations from the ideal diffusion-controlled crystal growth
in fluid cells under microgravity as crystals have been
observed to move inside the fluid cells (Chayen et al., 1997,
Lorber et al., 2000).

Although the space platforms are under microgravity, the
fluid cells still experience forces due to accelerations that
alter the nearly weightless condition (Ostrach, 1982). The
predominant component of theses forces occurs from
random accelerations called g-jitter (Knabe and Eilers,
1982). The unexpected movements of the crystals observed
in space experiments can be related to g-jitter and can
adversely affect the growth and quality of crystals (Chayen
et al., 1997, Lorber et al., 2000, Kawaji et al., 2003,
Simic-Stefani et al., 2006). In an orbiting spacecraft, g-jitter
occurs in random directions with different frequencies and
acceleration levels. These accelerations are produced by
mechanical vibrations caused by the operation of equipment
such as pumps, fans and motors, crew motion, spacecraft

Many studies have been conducted on the motion of a
particle in the past that confirm the vibrations can induce a
hydrodynamic force on a particle suspended in a fluid of
different density. Stokes (1851), Boussinesq (1885), and
Basset (1888) derived expressions for the forces affecting a
particle submerged in a fluid subjected to harmonic and
arbitrary motions. More recently, an equation of motion for
a small rigid sphere in a non-uniform flow was developed
by Maxey and Riley (1982). Chelomey (1983) studied the
behavior of solid particles immersed in a liquid-filled
container subjected to vibrations and found that particles
heavier than the surrounding fluid would emerge at the top
while particles lighter than the fluid would move
downwards. He attributed this paradoxical behavior to the
effects of non-uniform forces developed during the
vibrations. Coimbra and Rangel (2001) developed an
analytical model for the periodic motion of a small particle
in a viscous fluid to analyze the motion of particles induced
by vibrations. In their analytical model they combined
virtual mass, Stokes drag, and history forces. Later, their
analysis was confirmed experimentally by Coimbra et al.
(2004) and L'Esperance et al. (2005).

Recently, Hassan et al. (2006a, 2006b) and Hassan and
Kawaji (2007a, 2007b) investigated the vibration-induced
motions of a particle in a fluid cell for different density
particles in water. Under the vibration conditions they
examined, water could be considered inviscid. Moreover,
Hassan et al. (2006c) studied both analytically and

experimentally, the effects of vibrations of a fluid cell on the
motion of a particle suspended by a wire in a fluid cell
under normal gravity. Their study showed the occurrence of
an attraction force on the particle when the particle
oscillated in an inviscid fluid near the cell wall. To mimic
the vibrations that produce g-jitter, the cell was oscillating
with a sinusoidal motion of known amplitude and
frequency.

When the particle was oscillating far away from the cell
walls, the vibrations had no effect on the mean particle
position. However, placing the particle close to one of the
cell walls resulted in a clear shift of the particle's mean
position towards that wall. They studied oscillations both
normal and parallel to the nearest wall. For both cases the
drift in the mean particle position increased with the cell
vibration frequency and amplitude and decreased with an
increase in the distance between the particle and the nearest
wall. Hassan et al. (2006c) also obtained analytical relations
for the shift in the mean position of the particle towards the
nearest cell wall when the particle was oscillating either in
parallel with the nearest cell wall or normal to it.

For example, for a particle oscillating in parallel with the
nearest cell wall, the drift towards the wall, XD is

XD 9 L a2 2 P PL L
32 g (2ps +P)2 H(1
which is the drift towards the wall for the particle oscillating
in parallel with it. In Equation (1), L is the distance from the
suspension point to the center of mass of the particle, g is
the gravitational acceleration, a is the cell amplitude, m is
the angular frequency (= 27f, where f is the frequency in
Hz), ps and pL are particle and liquid densities
respectively, Ro is the particle radius and H is the initial
distance between the stationary particle and the nearest cell
wall. Figure 1 shows the experimental data and a plot of
Equation (1) for the particle drift versus cell vibration
frequency for a steel particle in water. Unfortunately,
Hassan et al. (2006c) did not include any investigations into
the mechanisms responsible for the attraction of the particle
to the nearest wall.

Recently, Simic-Stefani et al. (2006) conducted direct
numerical simulations to investigate the effect of small
vibrations on particles such as protein crystals growing in
protein crystal growth (PCG) cells under zero gravity. Their
study revealed the effect of six parameters on the crystal
motion; frequency and amplitude of the vibration, crystal
diameter and density, and fluid viscosity and density.
Moreover, they observed formation of a vortex ring around
a moving particle that changed the direction of its rotation
based on the particle-to-fluid density ratio. However,
Simic-Stefani et al. (2006) did not investigate the effect of
the nearest wall on the particle oscillation in their study.

The objective of this study is to numerically investigate the
mechanisms that cause attraction of the particle to the
nearest wall in a fluid cell. To achieve this objective, direct
numerical simulations were conducted for a particle in a
fluid cell under zero gravity. The fluid cell was subjected to
sinusoidal vibrations of known amplitude and frequency

a cell amplitude (m)
A cell vibration amplitude
f frequency (Hz)
f body force per unit mass
F hydrodynamic force
g gravitational acceleration (ms-2)
G external body force
gs steady residual acceleration
H initial distance between the stationary particle
and the nearest cell wall
I moment of inertia matrix
L distance from the suspension point to the center
of mass of the particle (m)
n unit normal vector on the surface of the particle
pointing into the particle
Ro particle radius (m)
t time (s)
T, moment acting on the particles
u velocity of fluid
i test function for the fluid velocity
V translational velocity
V velocity
XD particle drift towards the cell wall (m)
X position of the center of mass of the sphere
x position of a given point on the surface of the
particle.

The Direct Numerical Simulation code, Partflow3d,
developed by Hu et al. (2001) is used in this work to
simulate the motion of a solid particle in a fluid cell. The
description of the code given below is mainly taken from
Hu et al. (2001) which the interested readers are referred to
for more detailed information. The code is based on a finite
element technique. Since the movements of the solid
particles and surrounding fluid are coupled, the numerical
code solves both the fluid and particle governing equations
combined into a single equation. The code considers the
motion of N rigid solid particles in an incompressible fluid.
It also considers a fluid domain, d20(t), at a given time t,
and domain of particles d, (t), where the index i (= 1, 2,
..., N) represents different solid particles.

Partflow3d uses a mesh composed of unstructured
tetrahedron elements that is suitable for the irregularity and
movement of the domain. The nodes on the particle surface
are moving with the particle and therefore the code solves a
moving boundary problem. To accommodate this issue, the
mesh is regenerated at each time step. The code is written in
3-D and is based on an Arbitrary Lagrangian-Eulerian
technique, which uses the standard Galerkin finite element
method. At each time step, the positions of the particle and
the mesh are updated explicitly, while the velocities of the
particle and fluid are determined implicitly.

Governing Equations

The governing equations for the motion of the fluid are the
conservation of mass,
Vu = 0, (2)
and the conservation of momentum,
Du
p/- = p/f+V.o, (3)

where u is the velocity vector, pf is the density of the fluid,
f is the body force per unit mass, a is the stress tensor
Du
and is given by,
Dt
Du cu
D + (u.V)u. (4)
Dt O.
For the rigid particles, the governing equations are
Newton's second law for the translational motion,
dV
m, V =G,+F,=G,- -a.ndS, (5)
at, (t)
and the Euler equations for the rotation,

d
dt

I, -+,) XI x,)
dt- ( ,

T,- (x-X,)x(a-n)dS,

S0,(t)
where m, is the mass, and I, is the moment of inertia matrix
of the ith particle, V, is the translational velocity of the
particles, o, is the angular velocity of the particles, G, is
the external body force, n is the unit normal vector on the
surface of the particle pointing into the particle, X, is the
particle's centroid and d.2 (t) denotes the boundaries of
the particles domain. Equations (5) and (6) also show that
the hydrodynamic force F, and the moment acting on the
particles T, are obtained by integrating the fluid stress over
the particles' surface.

In the specific use of Partflow3d for this work, a sinusoidal
motion is applied to the fluid cell containing the particle.
The position and the acceleration of the fluid cell are
sinusoidal functions of time in a fixed frame of reference.
The cell acceleration therefore, can be specified through the
body force f which can be defined as,
f =g Awt sin(ot)n, (7)
where g, represents steady residual accelerations, A is the
cell vibration amplitude (half of peak-to-peak amplitude),
and co is the angular frequency (=22iJ). The residual
acceleration was assumed to be zero in this work.

The non-slip boundary condition on the particle surface was
considered as,
u = + o,(x- X,), (8)
where u is the velocity of fluid, V, is the velocity of particle,
X, is position of the center of mass of the sphere, and x is
the position of a given point on the surface of the particle.
The non-slip boundary condition for the boundaries of the
computational domain (cell walls) was
u = 0. (9)

Combined fluid-solid formulation

In order to use the finite-element method, Hu et al. (2001)
derived a weak formulation that incorporates both the fluid
and particle equations of motion, Equations (3), (5), and (6).
This was achieved by multiplying Equation (3) by the test
function for the fluid velocity, u and integrating over the
fluid domain at a time instant t. The result is

S Dt f ud& + f ( Vii)d2

f J( n)idS = 0. (10)
I
For the particles, the test function will be replaced by
Equation (8). By using equations of motion for the particles,
Equations (5) and (6), the following relation would be
derived
- (a n)idS
02

+ V, G,I
I I IN dt
+ d(I) 0. (12)
I<
Similarly, the weak formulation for the mass conservation,
Equation (2), can also be obtained by multiplying its
corresponding test function and integrating over the fluid
domain.

As it is clear from Equation (12), for the fluid-particle
system, the hydrodynamic forces and moments acting on the
particle do not explicitly appear in the formulation. This is
due to considering the fluid and particles as one system
which turns the hydrodynamic forces to internal forces. This
reduces the computational time, and is an advantage when it
comes to compute the fluid and particles' behaviors
numerically.

Results and Discussion

In this work, the vibration-induced motion of a single
spherical particle in a liquid-filled cell was considered. The
particle is initially placed near one of the vertical cell walls
as shown in Figure 2. As the cell is vibrated in the
z-direction with a specified amplitude and frequency, the
particle would be induced to vibrate in the z-direction,
parallel to the nearest wall, due to a density difference
between the particle and surrounding liquid. It is important
to note that the particle was fixed in y-direction in the
simulations in order to maintain an average for the
hydrodynamic force. This helped better studying the force
acting on the particle by preventing the particle from
moving towards the wall.

For the present 3-dimensional simulations, a fluid cell with
internal dimensions of 140 mm height and 50 mm x 50 mm
cross section under zero gravity environment was
considered. A spherical stainless steel particle with a
diameter of 12.7 mm was considered to be placed very close
to one of the cell walls (Figure 2) with its centroid at x = 7.6
cm, y = 4.3 cm, and z = 2.5 cm, so that the distance between
the edge of the particle and the nearest cell wall was 650
microns.

To understand the effect of the nearest wall on the
hydrodynamic forces acting on the particle vibrating near a
fluid cell wall, direct numerical simulations were conducted

for a liquid with a viscosity of 1 cSt and density of 1,000
kg/m3. The fluid cell was subjected to sinusoidal vibrations
of 1.0 mm amplitude (half of peak-to-peak) and two
different frequencies of 7 and 15 Hz. All the numerical
results presented in this work were obtained with a
sufficiently small mesh that the results would not vary more
than 1% with further mesh refinement.

All the simulations were conducted in 3-D, and the
predicted velocity and pressure fields extracted from the
horizontal z-y plane that cuts through the centroid of the

Figure 2 Schematic of the fluid cell and particle showing
the horizontal plane for flow visualization.

particle as shown in Figures 2 and 3 will be presented below
to reveal the flow pattern around the particle.

The flow pattern around the moving particle will be
described according to the diagram shown in Figure 3.
There are four different zones of interest close to the
particle: the front and back zones which lie in front and
back of the moving particle, the wall zone between the
particle and the nearest wall, and the fluid zone on the
opposite side.
------

Particle Back Zone
Fluid ZoneWall Zone
--- Front Zone

Particle Movement Direction
Cell Wall

Figure 3 A view of the particle and fluid motion in the
horizontal plane passing through the centroid of the particle.

C""C~,,

cLo

In one cycle, the particle moves from z = +A to -A and back
to +A, where A is the particle amplitude. To discuss the
results of simulations of the particle movement in the fluid
cell, let us consider five different positions of the particle in
each cycle:

Position 1: particle is moving in -z direction with its
maximum velocity,
Position 2: particle is half-way towards its end position (z
= A/2),
Position 3: particle reaches the end position (z = -A) and
its velocity is zero,
Position 4: particle is moving in +z direction and is
half-way to its maximum velocity,
Position 5: particle is moving in +z direction and reaches
its maximum velocity.

Vibration-Induced Particle and Fluid Motion

In the following figures, the units of velocity and pressure
are cm/s and g/cm.s2, respectively. The flow patterns
predicted are discussed below for the particle positions 1 to
5 for the cell vibration amplitude of 1.0 mm and two
different vibration frequencies of 15 Hz and 7 Hz.

Figures 4 and 5 show the particle at Position 1 for cell
vibration frequencies of 15 and 7 Hz, respectively. As it is
clear from these figures, the pressure in the wall zone and
the fluid zone are lower than the other areas around the
particle for both cases. However, the low pressure area
around the wall zone is larger than the fluid zone and also
the fluid velocity is the largest in the wall zone. This clearly
indicates that the particle would be attracted towards the
nearest wall based on the Bernoulli's principle.

- N,
-N
.---

-20
*40
40
S* 1 i20
-140

Figure 4 Position 1: Velocity and pressure distributions
around the particle for cell amplitude of 1.0 mm and
frequency of 15 Hz.

Another interesting observation from these figures is that
the flow direction in the wall zone is opposite to the
direction of the moving particle. This interesting
phenomenon occurs possibly due to the low viscosity of the
fluid. As the particle moves in a given direction, the liquid
between the particle and the nearest wall is squeezed and
directed towards the opposite direction. If the fluid were

highly viscous, the shear stress would be much greater and
the liquid may not be forced to flow in the direction
opposite to the particle.

Figure 5 Position 1: Velocity and pressure distributions
around the particle for cell amplitude of 1.0 mm and
frequency of 7 Hz.

Figures 6 and 7 show the particle at Position 2 for cell
vibration frequencies of 15 and 7 Hz, respectively. As
shown, the pressures in the front zone and the wall zone are
still lower than the other parts of the fluid which again
indicates the particle attraction towards the nearest wall.

p
140
120
100
8o

680
100

-12
-40
-120
-110

Figure 6 Position 2: Velocity and pressure distributions
around the particle for cell amplitude of 1.0 mm and
frequency of 15 Hz.

At Position 3 (Figs. 8 and 9), the particle stops and changes
its direction. Because the fluid is still moving in the -z
direction, the pressure in the back zone becomes higher than
in the front zone. When the particle reverses its direction
and moves in the +z direction (Position 4 as shown in
Figures 10 and 11), the pressure in the front zone is still
higher than in the back zone (considering that the back and
front zones are now switched). However, the pressure in the
wall zone is still low and at the same time the flow velocity
in the wall zone is the highest in the fluid. This suggests that
the particle should still be attracted to the wall.

Figure 7 Position 2: Velocity and pressure distributions
around the particle for cell amplitude of 1.0 mm and
frequency of 7 Hz.

z

p,
leo

120
1o0

80
20

-160
-s

Figure 8 Position 3: Velocity and pressure distributions
around the particle for cell amplitude of 1.0 mm and
frequency of 15 Hz.

20
80
-20

-20

Figure 9 Position 3: Velocity and pressure distributions
around the particle for cell amplitude of 1.0 mm and
frequency of 7 Hz.

p ,~

Ii

Figure 10 Position 4: '
around the particle for
frequency of 15 Hz.

Z
LY
p
140
120
100
so
80
60
40
20
0
-20
-40
-80
-100
-120
-140
-1 -1

velocity and pressure distributions
cell amplitude of 1.0 mm and

. V

.4

L,
p
160
120
n
so
00
6O
20
0
-2a
40
-60
-80
-100
-120
-140
-160

Figure 11 Position 4: Velocity and pressure distributions
around the particle for cell amplitude of 1.0 mm and
frequency of 7 Hz.

Finally, at Position 5 (Figures 12 and 13) when the particle
reaches its highest velocity, the wall zone still has the
lowest pressure and the highest velocity in the fluid. At the
same time, the pressure in the front zone is lower than in the
back zone. The pressure and the velocity predictions again
suggest that the particle should be attracted to the wall.

Studying the flow pattern around the particle for a 1 cSt
fluid suggests that the attraction force on a particle
oscillating in the vicinity of a wall in a fluid cell is governed
by the Bernoulli's principle. The fluid cell vibration induces
the particle motion which in turn results in the fluid motion.
The velocity and pressure in the gap between the oscillating
particle and the nearest wall reach the highest and lowest
values, respectively, compared to those in other regions.
Consequently, the particle is attracted towards the nearest
wall, as theoretically predicted by Hassan et al. (2006c) for
an inviscid fluid-filled cell.

Figure 12 Position 5: Velocity and pressure distributions
around the particle for the cell amplitude of 1 mm and
frequency of 15 Hz.

Figure 13 Position 5: Velocity and pressure distributions
around the particle for cell amplitude of 1.0 mm and
frequency of 7 Hz.

Attraction force exerted on the particle

The simulation results point to the existence of an attraction
force on the particle towards the nearest cell wall for a cell
filled with a 1 cSt fluid. Figures 14 and 15 show the
hydrodynamic forces exerted on the particle in the
y-direction for a cell vibration amplitude of 1.0 mm and
frequencies of 7 and 15 Hz, respectively. As these figures
show, an attraction force on the particle towards the nearest
wall is generated. The averaged force over one period of
oscillation is calculated to be equal to 1.97 dynes for 7 Hz
and 9.89 dynes for 15 Hz frequency.

The main contribution to the attraction force is related to the
maximum velocity of the particle. Note that the negative
velocities in Figures 14 and 15 indicate the particle moving
in -z direction. Therefore, both the positive and negative
peaks in the particle velocity in both figures correspond to
the maximum magnitude of the particle velocity. One can
clearly note from Figures 14 and 15 that the maximum
attraction force occurs when the particle velocity reaches a

Figure 15 Hydrodynamic force exerted on the particle in
y-direction and particle velocity in the direction of cell
vibration for cell vibration amplitude of 1.0 mm and
frequency of 15 Hz.

Conclusions

Direct numerical simulations on a spherical particle
suspended in a fluid cell were performed in order to
understand the mechanisms leading to attraction of the
particle to the closest wall in a fluid cell. The particle's
motion was induced by small sinusoidal vibrations of
known amplitude and frequency under zero gravity
environment. The focus of the work was on the
hydrodynamic forces affecting the particle. The simulations,
performed for a fluid of 1 cSt viscosity and 1000 kg/m3
density, revealed an attraction force towards the nearest cell
wall for this fluid.

Studying the pressure distribution around the particle
showed that for the 1 cSt fluid, the pressure between the
particle and the nearest wall is lower than in other regions
inside the fluid cell. Moreover, the velocity distributions
showed that for this fluid, the highest velocity in the fluid
cell occurs between the particle and the nearest wall. Based
on these observations, it is clear that the hydrodynamic
attractive force on the particle is related to the Bernoulli's
principle of reduced pressure in high velocity zones.

Acknowledgements

The authors would like to thank the Canadian Space Agency
for an SSEP grant to financially support this work.